Question

If a $\Delta \,ABC$ has vertices $A (-1,7), B (-7,1)$ and $C (5,-5)$, then its orthocentre has coordinates:

• A. $(3,-3)$
• B. $\left(-\frac{3}{5}, \frac{3}{5}\right)$
• C. $(-3,3)$
• D. $\left(\frac{3}{5},-\frac{3}{5}\right)$

Answer 1

Answer: (C)

Let orthocentre is $H \left( x _{0}, y _{0}\right)$

$\Rightarrow \left(\frac{y_{0}-7}{x_{0}+1}\right)\left(\frac{1+5}{-7-5}\right)=-1$
$\Rightarrow 2 x _{0}- y _{0}+9=0$ .....(1)
and $m_{BH} \cdot m_{AC}=-1$
$\Rightarrow \left(\frac{y_{0}-1}{x_{0}+7}\right)\left(\frac{7-(-5)}{-1-5}\right)=-1 $
$\Rightarrow x_{0}-2 y_{0}+9-0$.....(2)
Solving equation (1) and (2) we get
$\left(x_{0}, y_{0}\right) \equiv (-3, 3)$